Non-Diophantine arithmetic as the mathematical foundation for quantum field theory

TL;DR

This paper introduces a novel non-Diophantine arithmetic framework to address the divergence issues in quantum field theory, providing a new mathematical foundation that yields convergent integrals for Feynman diagrams.

Contribution

It develops new non-Diophantine arithmetics and demonstrates their application in making divergent QFT integrals convergent, offering an alternative to traditional renormalization.

Findings

Non-Diophantine integrals are convergent where traditional ones diverge.

New properties of non-Diophantine arithmetics are established.

Potential for a rigorous mathematical foundation in QFT using non-Diophantine arithmetic.

Abstract

The problem of infinities in quantum field theory (QRT) is a long standing problem in physics.For solving this problem, different renormalization techniques have been suggested but the problem still persists. Here we suggest another approachto the elimination of infinities in QFT, which is based on non-Diophantine arithmetics - a novel mathematical area that already found useful applications in physics. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. This allows using these arithmetics for computing integrals describing Feynman diagrams. Although in the conventional QFT these integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined.